Sensitivity and identifiability analysis of a conceptual-lumped model in the headwaters of the Benue River Basin, Cameroon: implications for uncertainty quantification and parameter optimization

Hydrological models are considered useful and valuable tools in several aspects of water resource management. They are increasingly being employed for simulating hydrological processes (Budhathoki et al. 2023; Velásquez et al. 2023), forecasting extreme events like floods and droughts (Ich et al. 2022; Moore & Cole 2022), analyzing future climate change and land use scenarios on available water resources and hydropower potential (Nonki et al. 2019, 2021b; Obahoundje et al. 2021; Rahvareh et al. 2023). For the larger number of existing hydrological models that vary from physical-based to conceptual models, the choice between the distributed, semi-distributed, and lumped-conceptual is very important for catchment hydrology. Given the expense of semi-distributed and distributed models in terms of input data and computational resources, many hydrological applications employ conceptual-lumped rainfall-runoff models to support water resource management techniques. Their ability to work with minimal data and provide enough credible information means they are a useful tool in many data-poor domains (Tegegne et al. 2017; Nonki et al. 2021c). All models are an imperfect simplification of the physical process and therefore have an inherent uncertainty associated with them. In data-scarce regions, where accurate input data are rarely available, the model uncertainty is compounded by input data uncertainties. Therefore, for science-based decision-making, an evaluation of uncertainty sources in the model is necessary for improving the structure of the models and lowering uncertainties (Refsgaard et al. 2006, 2007). This research topic was one of the three major objectives of the International Association of Hydrological Sciences (IAHS) Panta Rhei Science Decade 2013–2022 (Montanari et al. 2013) and constitutes one of the unsolved problems in hydrology (Bloschl et al. 2019).

Sensitivity and identifiability analyses are now invaluable strategies for model parameterization, calibration, and optimization, as well as uncertainty quantification and reduction (Saltelli et al. 2006; Guse et al. 2020; Nonki et al. 2021c). The former shows how errors in input data can affect model simulations (Saltelli 2002), even as the latter expresses how well the parameter is defined in the model structure (Abebe et al. 2010). Although several studies have addressed this issue (search for Shin et al. (2015); Song et al. (2015); Pianosi et al. (2016) and Devak & Dhanya (2017) for a complete review) and new applications including data science and machine learning are being developed (Saltelli et al. 2021), there are still some research challenges in this area (Razavi et al. 2021). Clarifying the relationship and position of SA in quantifying uncertainty as well as enhancing the use of SA to assist decision-making are two of the six most important challenges highlighted through Razavi et al. (2021). In addition, the majority of SA studies are based on one or two objective functions. According to Guse et al. (2020), the use of multiple objective functions in SA and parameter identifiability studies is of paramount importance because this helps to identify the group of parameters that most influence each part of the hydrograph (e.g. low and high flows) as well as the entire hydrograph (Boyle et al. 2000; Wagener et al. 2003).

Several studies have addressed this information gap (Li et al. 2021; Liang et al. 2021; Singh & Jha 2021; Tibangayuka et al. 2022). Singh & Jha (2021) investigated the impact of catchment size and simulation time steps in performing SA. They observed that the sensitivity of some parameters does not depend on watershed size and simulation time step, while other parameters are sensitive for small- and medium-sized watersheds but not for large watersheds at a daily time step. Li et al. (2021) investigated the impact of SA on parameter optimization. The case study was conducted in four watersheds in China by using the Soil and Water Assessment Tool (SWAT) model and Sensitive Parameter Combinations (SPCs). They found that no more than 10 sensitive parameters could be identified out of 27 modifiable parameters for each watershed, suggesting that sensitivity parameter optimization can greatly reduce the computational cost of SWAT streamflow simulations while ensuring their accuracy. Liang et al. (2021) explored the effect of sensitivity and uncertainty evaluation for discharge prediction in the Yalong River Basin (YLRB) of Southwest China using the SWAT model and three optimization algorithms. Their results indicated that some parameters could significantly affect the discharge in the study catchment, while complex parameter combinations reacted differently under the above-mentioned three optimization algorithms. Tibangayuka et al. (2022) quantified the implications of Hydrologiska Byrans Vattenavdelning (HBV) model parameter uncertainties through sensitivity and identifiability analyses in the Wami Ruvu Basin, Tanzania. They found that the parameter identifiability of the HBV model varies spatially in the basin, while parameter uncertainties significantly influence the model output. All these studies have proven and strongly recommended that a comprehensive parameter sensitivity and uncertainty analysis is a vital step in setting up any hydrological model to reduce the number of parameters while still addressing all relevant hydrological processes.

These studies are all context-specific; each catchment study has a relatively unique aggregate of geographical, climatic, geological and hydrological conditions. Therefore, the predictive ability of a rainfall-runoff model depends on its structure, the quality of the input data, both in terms of spatial and temporal resolution, and the experiment design and execution. The purpose of this work is to assess the applicability of a daily conceptual time-step rainfall-runoff model, HYdrological MODel (HYMOD), to the Headwaters Benue River Basin (HBRB) for future water resource management and policy. The study proposes using two approaches of SA to identify which model parameters most influence the model output and quantify how well the parameters are defined in the model structure using six performance criteria. In doing so, we (i) provide a plausible comparison between the two approaches (local and global) commonly used in the SA studies, (ii) contribute to understanding the impact of selected performance criteria in the sensitivity and identifiability analysis, and (iii) assess the role of the sensitivity and identifiability analysis on the uncertainty quantification, parameter optimization, and model performance. This paper is structured as follows: Section 2 describes the Materials and Methods; Section 3 presents the Results and Discussions, and Section 4 provides that a summary from conclusions is drawn.

The basin enjoys a Sudan-Sahelian climate (tropical humid climate), which is characterized by two distinct seasons: a dry season from November to April and a wet season from May to October. This is a unimodal rainfall zone with annual rainfall between 900 and 1,500 mm (Dassou et al. 2016), which gradually decreases from the south (Adamawa Plateau Highlands) to the north of the basin (Chad Plains). In contrast to rainfall, the temperature inside the basin increases gradually from south to north, with an average annual basin temperature of 28 °C. Vegetation of the area is dominated by savanna (59%), wooded savanna (38%), and highland meadow (3%). The Benue River watershed is defined by different classes of soil type: silty clay loam, silty clay, silty loam, silt, and sandy. The elevation varies from 220 to 2,260 m and is characterized by the Adamawa Plateau, and Alantika and Mandara mountains (Dassou et al. 2016).

Daily rainfall and potential evapotranspiration (PET) data were used as inputs to the hydrological model, while measured daily streamflow time-series were utilized for model calibration and validation. Daily rainfall registered at 25 weather meteorological stations (corresponding to 1 station/2,500 km²) in the catchment and neighboring areas and daily PET computed with the Penman formula were provided by the Direction of the National Meteorology of Cameroon (DNM). Figure 1 shows the geographical location of the meteorological stations, whereas Table 1 provides the station names, coordinates, altitudes, recording period, and data quality assessment.

Daily measured streamflow data for three gauging stations (Garoua, Riao, and Buffle Noir) that are located in the basin were obtained from the environmental information system for the water resource (SIEREM) database (Boyer et al. 2008; http://hydrosciences.fr/sierem). These three gauging stations are considered here as sub-catchments, and Table 2 shows the physiographic and hydrological characteristics of these gauging stations.

Two SA approaches were used in this research: the local SA (LSA) approach, which helped to identify the parameters that individually most influence the model outputs, and the global SA (GSA) approach, which considers the interactions between the model parameters and helps to refine the behavioral/non-behavioral ranges for a parameter and perform the identifiability analysis.

For the individual SA, a widely applied and recommended one-factor-at-time (OAT) method is used (Abebe et al. 2010; Pianosi et al. 2016). The optimum model parameter set was first obtained during the initial model calibration using multiple objective functions. For each model parameter that varies for its initial range (see Table 2), 2,000 values were generated using the uniform distribution function, and the model was therefore run while keeping other parameters at their optimized values. For each simulation, six performance criteria, namely Nash and Sutcliffe efficiency (NSE; Nash & Sutcliffe 1970), percent bias (PBIAS; Zhang et al. 2011), Pearson correlation coefficient (r; Moriasi et al. 2007), standardized root mean square error (RSR; Moriasi et al. 2007), Kling–Gupta efficiency (KGE; Gupta et al. 2009), and composite function of KGE and inverse of KGE (OF; Lemaitre-Basset et al. 2021) were computed. These criteria were selected based on the connection between them and part of the hydrograph, as well as the hydrological components that they consider (see Table 3 for the names, formulations, and the threshold value of these criteria for behavioral simulations). The parameter is identified as influencing or not influencing the model outputs if changes in its value influence the performance criteria or not.

For the GSA and parameter identifiability analysis, we implemented the Monte Carlo approach. It is a GSA method widely used in SA studies (Sobol’ 2001; Saltelli 2002; Abebe et al. 2010) and implemented in many algorithms of SA (Beven & Freer 2001; Sobol’ & Myshetskaya 2008; Azzini et al. 2021). This method has the advantage of implicitly accounting for the interactions between model parameters. It is based on running many simulations of the model using a large random sample of input variables. Considering the initial ranges of different parameters, we generated 50,000 model parameter sets. The model was run for each model parameter set with the model parameters sampled simultaneously. The simulations, as well as model parameters, were split into behavioral and non-behavioral simulations/parameters based on the threshold value of each performance criterion mentioned above. The parameter is then said to be more precisely identified in the model structure, or more sensitive to the model output if the range of behavioral parameters is smaller than the initial range. This is assessed by comparing the posterior distribution of the behavioral parameter with the prior distribution of the same parameter, which is assumed here to be uniform (Quan et al. 2015). If the two distributions (prior and posterior) deviate significantly, the parameter is considered a sensitive parameter (Sun et al. 2012; Quan et al. 2015).

At this stage, the Monte Carlo optimization algorithm and the split-sample test (Klemeš 1986) were applied. The data time-series were divided into two sub-periods (calibration and validation) and the first year of each sub-period was considered as a warm-up period. The lower and upper bounds of each parameter obtained from the behavioral simulations were used to evaluate the descriptive capabilities of the model in the calibration phase using the multi-objective functions mentioned above, keeping constant the values of the parameters identified as not influencing the model output and poorly defined within the model structure. This can help to reduce the parameter range and space and thus equifinality (Cibin et al. 2014). The optimum model parameter sets obtained from 50,000 model runs were then used to check the predictive capacities of the model in the validation period. The performance of the model was then evaluated using graphical analysis (visual hydrograph comparison as well as flow duration curves) and statistical criteria listed in Table 4. The behavioral parameter sets obtained by constraining the model parameters into a small range with respect to the KGE were used to simulate the discharge during the validation period and the uncertainty in the model prediction was assessed by plotting the uncertainty band. The P-factor and the R-factor were also used to quantify the proportion of the measured discharge that falls inside the uncertainty band and to represent the average width of the given uncertainty limits divided by the standard deviation of the observations, respectively. Ideally, most of the measured discharge should fall within the uncertainty band (P-factor →1) while having the narrowest band (R-factor → 0) (Abbaspour et al. 2009).

Table 4

Mathematical formulation and optimal as well as threshold values of each objective measure for acceptable simulations

Objective criteria .	Formulation .	Optimal value .	Threshold for behavioral simulations .
NSE (Q)		1
RSR		0
PBIAS		0
r		1
KGE (Q)		1
OF		1

Objective criteria .	Formulation .	Optimal value .	Threshold for behavioral simulations .
NSE (Q)		1
RSR		0
PBIAS		0
r		1
KGE (Q)		1
OF		1

and represent, respectively, the measured and modeled streamflow in the time step i; and represent their mean values, and n is the total number of time steps of simulation. α is the ratio between the standard deviations of modeled and measured streamflow, while β is the ratio between their mean values.

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We also notice that the residence time of the quick-flow reservoir parameter (RF) is sensitive with regard to RSR and r criteria since this parameter controls both the timing and shape of the hydrograph and therefore has little effect on high-flow series (NSE) and no influence on the volume error (PBIAS). This result underscores the importance of using several objective functions for the SA, as the group of sensitive parameters is related to the selected objective function. This finding is consistent with other research (Abebe et al. 2010; Zelelew & Alfredsen 2012; Guse et al. 2020). The results also reveal that the parameter sensitivity increases with the increasing catchment size.

For all three parameters (Alpha, RS, and RF), the parameter related to slow reservoirs (KS) has the highest sensitivity, followed by the parameter associated with fast flow (KF), and then the parameter related to water partitioning between fast and slow reservoirs (Alpha) as measured by KGE and OF criteria. The above shows that in the HYMOD, slow response processes become more important than fast-flow generation processes (Parra et al. 2018). This is consistent with the results of the identifiability analysis in which, for the parameter Alpha, there is no improvement of parameter range by contrasting the majority of performance criteria compared to RF and RS, which was precisely constrained by contrasting some performance criteria (see Figure 5). The parameter sensitivity and identifiability increase with increasing catchment size as found with LSA. This means that lumped-conceptual models are better suited to large catchments than to small ones. This can be explained by the fact that the functional behavior of catchments differs considerably depending on the scale of the catchment, and that catchment size is one of the five most important explanatory variables influencing runoff simulations (Poncelet et al. 2017). Consequently, for large catchments with smooth hydrological behavior, it is easier for models to reproduce different hydrological processes.

Comparing the results of local SA with those of global SA shows that both give similar results in terms of parameter impact on the model output with regard to both objective functions and catchment size. However, we assess the GSA to have advantages over LSA as GSA not only provides the influential or non-influential model parameters to the model output, it is also used to constrain the model parameters in a small range that helps to reduce the equifinality and therefore quantify and reduce the uncertainties in the model output.

For the three sub-catchments considered, the calibration efficiency is higher than the validation efficiency with a loss in the model efficiency of 0.1, 0.07, 0.08, and 0.03 for the KGE, NSE, RSR, and r criteria, respectively (see the mean model efficiencies of the three sub-catchments in Table 5). This result is somewhat expected as the models are generally assumed to represent calibration data better than validation data. In addition, the model is calibrated to better represent the hydrological conditions of the catchment during the calibration period, which are never exactly the same during the model validation period (Merz et al. 2009).

The statistical evaluation of the model performance also shows that the model performance increases with increasing catchment size. This can be explained by the fact that the SA and the parameter identifiability analyses show that the precisely identified model parameters increase with the catchment size and, consistent with Guse et al. (2020), the precisely identified model parameters improve the model robustness and performance. This result is consistent with those of Merz et al. (2009) and Nonki et al. (2021c), who found that the performance of the conceptual rainfall-runoff model increases with catchment size. Catchment size is one of the five most important explanatory variables influencing runoff simulations (Poncelet et al. 2017). Therefore, it is expected that for large catchments with smooth hydrological behavior, it is easier for the models to breed streamflow. In addition, consistent with Poncelet et al. (2017), model performance decreases with precipitation and streamflow variability. For example, within the Buffle Noir outlet, the streamflow variability is more pronounced with a discharge coefficient of 39% compared to Riao (22%) and Garoua (15%) sub-catchments (see Figure 10). This may be due to the fact that this sub-catchment has a clear torrential character and each heavy rainfall will end in a definite peak flow.

The relative error-based statistical analysis (PBIAS) shows that the model underestimates the total discharge during the calibration and validation periods at the different gauging stations (see Table 5). This underestimation is more pronounced during the validation period than during the model calibration period because the model adjusts its parameters to the over- or underestimation of the input data during the calibration period (Nonki et al. 2021a). We also find that the model bias increases with the size of the catchment. This may be a consequence of the density and distribution of the rainfall stations considered for the calculation of mean areal rainfall (1/2,500 km²). Xu et al. (2013) found that increasing the number of rain gauges for the calculation of mean areal rainfall gradually reduces the range of simulated hydrographs and absolute errors, with a higher probability of over- or underestimation of peak flows when the number of rain gauges considered is smaller compared to a threshold number. Similar results have also been reported by Merz et al. (2009) and Xiaojun et al. (2021).

Conceptual rainfall-runoff models are widely used in many hydrological applications to support water resource management practices. They provide an advantage in data-poor regions due to their ability to use limited data and generate sufficiently reliable information. The main challenge with this type of hydrological model remains the flexibility to determine an optimal set of model parameters due to several sources of uncertainty. This study is being carried out in the HBRB in Cameroon and has the aim of developing a rainfall-runoff model that is appropriate in the context of the hydro-climatic characteristics of the basin. Local and global SA approaches were applied to identify which model parameters have the greatest impact on model output and how well model parameters are defined within the model structure using six performance criteria to reduce and predict model uncertainty and improve model performance. The results showed that the group of parameters sensitive to different hydrological components depended on the selected objective function using both local and global SA approaches. However, soil and evapotranspiration routine parameters (SM,max and Bexp) are sensitive to all the selected objective measures. We also found that the more precisely the model parameters are constrained within a small range, the smaller the model uncertainties and therefore the better the model performance. In addition, the best simulation versus measured discharge is within the narrow band of model prediction uncertainties and shows that the simulated discharge is in good agreement with the observations, indicating that the hydrological model performs well in this basin. The parameter sensitivity and identifiability, as well as the model performance, increase with the catchment size.

In summary, we have demonstrated the role and relationship between SA and uncertainty quantification and highlighted the importance of SA and parameter identifiability in uncertainty prediction and parameter optimization. Within this context, we conclude that the application of HYMOD to support various water management initiatives in this catchment is possible.

The authors gratefully acknowledge the data providers. The first author was supported by DAAD within the framework of the climapAfrica programme with funds from the Federal Ministry of Education and Research (grant no. 57610298). The authors gratefully acknowledge the constructive comments and valuable suggestions of the three anonymous reviewers and the Subject editor, which enormously improved the presentation of the final manuscript

Data cannot be made publicly available; readers should contact the corresponding author for details.

The authors declare there is no conflict.